The MANIAC, page 6
Cantor was born and raised in Russia, a nation whose inhabitants have become famous for their depth of feeling, the intensity of their religious and political beliefs, and a certain inclination for all things tragic, and while one could argue that these things are nothing but cultural clichés, easily discarded, in Cantor they all appeared to come alive, and they help in part to explain the complex and tortuous relationship he had with his own ideas. According to all accounts, he was a very pious Lutheran and an extremely sensitive soul. While he defended his theory in public, privately he could hardly deal with the consequences of his undeniably brilliant discovery, or what it appeared to be saying about the world. In some way, he confided to his daughter, he felt that his enormously expanded infinities put God into question. Or, at the very least, the outdated concept we had of Him and of His creation. These theological dialectics, in which he played the accuser and the accused, pained him as much as the vicious attacks that he received from so many of his peers. The great Henri Poincaré called his work “un beau cas pathologique,” a disease from which mathematics would eventually be cured, while others dismissed it as mathematical insanity, utter nonsense, or “no more than fog on a fog.” They went so far as to label him a scientific charlatan and a corrupter of youth. It was not only those insults that pained him but also the excessive admiration that his ideas garnered from others. “Not even God will expel us from the paradise which Cantor has created,” Hilbert wrote. Cantor, who should have felt supported and appreciated by the applause and recognition of someone of Hilbert’s stature, felt this praise unwelcome and unnecessary, because he was not working for fame or money, nor to put his name down in history. No, he was answering to a higher calling, one that was its own reward, something important enough to be an end in itself. Ever since he was a boy, Cantor had heard what he called “an unknown, secret voice” compelling him to study mathematics. He later became convinced that he had been able to develop his infinities through divine intervention. As with all true revelations, he believed that they would lead the human race toward a greater, transcendent certainty. But the exact opposite was taking place. No one could make head or tail of his infinities, and his opponents did everything they could to stunt his career and thwart his investigations. While undoubtedly worthy of a professorship in any of Prussia’s leading universities, Cantor languished in the backwaters of Halle, venting his frustration to a diminishing circle of friends and colleagues. “The fear my work inspires in some,” he wrote, “is a form of myopia that destroys the possibility of seeing what is really infinite, even though, in its highest and most perfect form, it is the infinite that has created and sustained us. I was recently shocked to receive a letter from Mittag-Leffler in which he writes, to my great amazement, that, after serious consideration, he regarded my intended publication as ‘approximately 100 years ahead of time.’ If so, I would have to wait until 1984, and that seems too much to ask of any man! No matter what they say, my theory stands as firm as a rock; every arrow directed against it will quickly return to wound the heart of its would-be assassin.” Trying to silence his opponents by making his theory complete, Cantor developed an entire hierarchy of infinities, but he struggled with increasingly strong episodes of uncontrollable mania, attacks that were followed by deep melancholy and the darkest possible depression. These spells became so regular that he was unable to work on mathematics; as a substitute, he devoted his boundless manic energy to try to prove that Shakespeare’s plays had actually been written by the English philosopher Francis Bacon, and that Christ was the natural son of Joseph of Arimathea, views that only helped to give more credence to the arguments of those who said that he was slowly going insane. In May 1884, he had a massive mental breakdown and had to be institutionalized at a sanatorium in Halle. According to his daughter, during these crises his entire personality would be transformed. He would shriek and yell uncontrollably at the doctors and nurses, and then fall stone-still and remain completely silent. One of his psychiatrists noted that, when he was not consumed by rage, he would give in to paranoid fantasies of persecution, imagining a devilish cabal that was working to undermine him and his work. During the periods between breakdowns, he continued to teach mathematics and kept toiling away at his infinities, but he was haunted by his own results, and became caught in a strange loop from which he could not extricate himself: he would first prove that the grand hypothesis that he was chasing after—the now infamous continuum hypothesis—was true, and then, a couple of months or even just weeks later, he would prove that it was false. This vicious cycle of truth and falsehood, truth and falsehood, truth and falsehood repeated again and again, compounding the misery that became the hallmark of his later years. Finally, on January 6, 1918, after suffering the death of his youngest son, numerous illnesses, bankruptcy, and severe malnutrition during the First World War, Cantor died from a heart attack in the Halle Nervenklinik, a university psychiatric institution where he had spent the last seven months of his life.
Cantor’s death was a tragedy for mathematics, but it did nothing to quell the arguments that his infinities had spawned. As the victim of an incomprehensible idea, he suffered for the great gift that he gave us, but he was certainly not the only one who agonized over the foundational crisis. In 1901, Bertrand Russell, one of Europe’s foremost logicians, discovered a fatal paradox in set theory, and it became a veritable obsession for him. It would not let him rest, even when he was sound asleep, because he would dream of it, again and again. To try to excise it, Russell and his colleague Alfred North Whitehead wrote a massive treatise intended to reduce all of mathematics to logic. They did not use axioms as Hilbert and von Neumann did, but an extreme form of logicism: to them, the foundation of mathematics had to be logical, and so they went about it, building mathematics from the ground up. This was not an easy task by any measure: the first seven hundred and sixty-two pages of their gargantuan treatise—Principia Mathematica—were dedicated solely to proving that one plus one equals two, at which point the authors dryly note, “The above proposition is occasionally useful.” Russell’s attempt to establish all of mathematics on logic also failed, and his paradox dreams were replaced by a new and recurring nightmare, which expressed the insecurities he felt regarding the value of his own work: In his reverie, Russell would stride along the halls of an endless library, with spiral staircases winding down into the abyss, and a high vaulted ceiling that rose up to meet the heavens. From where he was standing, he could see a young, gaunt librarian pacing the rows of books with a metal pail, such as one would use to draw water from a well, hanging from his arm, an undying fire burning within it. One by one, he would pick up the volumes from the shelves, open their dust-covered jackets, and flip through their pages, placing them back or tossing them into the bucket, to be consumed by the flames. Russell would watch him advance, knowing, with that certainty that we can only fully experience in dreams, that the young man was edging toward the last extant copy of his Principia Mathematica. When he picked it up, Russell would strain to try to make out the expression on the man’s face as he leafed through his book. Was that the beginning of a smile he could see creeping over his features? Was it disgust? Boredom, perhaps? Confusion? Disdain? The young man would put down the bucket, its flames licking at his fingertips, and stay there, unmoving, holding the book with both of his hands, his muscles taut from its excessive weight, and then he would suddenly look at the old logician, who would wake up in his bed, screaming, not knowing the fate of his work.
Russell and Whitehead covered more than two thousand pages with dense notations and obscure logical schemes to try to create a consistent and complete foundation for mathematics, while von Neumann’s doctoral thesis was so concise, his set of axioms could be written down on a single sheet of paper. Although it later turned out that his attempt was also unsuccessful, his audacity and succinctness did not go unnoticed, and he soon became famous for it among his peers. His thesis was an early demonstration of the style that he applied to all of his later work: he would pounce on a subject, strip it down to its bare axioms, and turn whatever he was analyzing into a problem of pure logic. This otherworldly capacity to see into the heart of things, or—if viewed from its opposing angle—this characteristic shortsightedness, which allowed him to think in nothing but fundamentals, was not merely the key to his particular genius but also the explanation for his almost childlike moral blindness.
Gábor Szegő
A god-shaped hole
He was a little devil, that one, but also an angel to those of us who saw the madness that was coming and fled from Germany before it was too late. I’m glad that I taught him when he was only a boychik, because he was very different as a grown man. A mathematical behemoth, yes indeed, but Hashem knows he was also a fool, and a dangerous one at that! Such a contradiction. It was like talking with two different people at the same time. Brilliant but childish, insightful yet incredibly shallow. Always gossiping that one. Always drinking! Wasting his time with idiotic friends, spending his money on high-priced whores, and morbidly interested in the stupidest things imaginable, like what cousin this or that baron had married, and how many legitimate and illegitimate children they had. I never understood his need for all that farkakteh small talk. He once spent half an hour explaining the many advantages of having a small Pekinese versus a Great Dane, and he was still going on about it when I got up and left. This, from the same person who made countless contributions to group, ergodic, and operator theory, and published thirty-two major papers in less than three years. But I came to him for help, nonetheless, because I knew that he was someone you could count on for important things. Janos was only twenty-seven and already a full professor in Princeton, constantly traveling back and forth between America and Budapest, Göttingen and Berlin. I was teaching mathematics at the University of Königsberg and living very comfortably, but I had learned enough from the two years of White Terror in Hungary, after Béla Kun’s short-lived communist government, when so many of my people were executed, hanged in public, tortured, imprisoned, or raped, just because some of the communist leaders had been Jewish, to know what to expect when the Nazis began to speak openly against us. I wrote to him, and we agreed to meet in Berlin, where I hoped to ask him if he could use his contacts so that my family and I could migrate to America. And where do you think he invited me to lunch? To Horcher, no less! The lavish, wood-paneled restaurant where the senior members of the Nazi Party would dine on crab to celebrate the political purge and butchery of the Night of the Long Knives just four years later. That was quintessential von Neumann. So accustomed to privilege that nothing but the best would do. Horcher was the finest restaurant in Berlin, so naturally that was where we should eat. When I arrived, I felt completely underdressed and out of place, but as soon as I mentioned his name, I was treated with the utmost respect and taken to one of the best tables in the house, where Janos was waiting for me smoking a thick cigar, illuminated by the autumn glow that shone through a pair of tall windows veiled by intricately woven lace curtains. Even if the Nazis only had a couple of seats in parliament at the time, I can’t help but shiver at the memory of the two of us sitting in that lion’s den at the corner of Lutherstrasse—the same spot that would become Himmler’s favorite place to dine—making escape plans while completely surrounded by royals, diplomats, spies, movie stars, politicians, moguls, and other wretched members of the German aristocracy.
We spoke about mathematics first, of course, and I could immediately sense the perverse influence that arrogant nudnik Hilbert was having on my former pupil. Janos had developed a strange compulsion, the same single-minded obsession with logic and formal systems that I have seen eat away at other great men. I was taken aback at how completely overzealous he had become. But then again, fanaticism was the norm in Mitteleuropa, even among us mathematicians. When he told me that he was close to realizing his dream of capturing the essence of mathematics in axioms that were consistent, complete, and totally free from contradictions, I scoffed. You should grow like an onion, I said, with your head in the ground! How could such a strict reduction—which creates as many problems as it solves—guide humanity toward anything resembling the paradise of rigorous clarity that he was dreaming of? And what kind of an Eden would that be? I wondered. Surely not a place where plants and trees could grow. He kept on ordering more food and drink, not even noticing that I did not touch mine. I told him that he should stay in America, and never return to Germany, but there was no way to convince him. He said he was working on something very important. He could feel an idea taking shape in the back of his mind, and feared that if he could not interact with Hilbert and other members of the Göttingen circle, he might lose it. I told him that it was better to lose an idea than to lose his life, but he looked at me in a way that made it clear that he saw it the other way round. Did I not know what was happening in quantum physics? It is all numbers, he said to me, these things do not behave like particles, they are not like bundles of matter or energy . . . They behave like numbers! And who better than us to comprehend that new reality? For Janos, it was not chemistry, industry, or politics that would shape the future of our world. It was mathematics. And that was why we needed to understand it to the deepest level. I had no way of imagining what he was talking about, because the tools and technology that he would later develop did not exist yet, but he became so uncharacteristically serious that, for reasons I cannot fathom, I began to shiver as if a gust of cold air had blown in from the outside. I believe he noticed it, because he immediately changed the subject and told me that I did not have to worry. America would welcome us with open arms. He spoke with such absolute confidence that I felt immediately relieved, but when I asked him for details, wanting to know to whom he would talk, and where he thought I might end up, he simply made a toast in my name and said that he would take care of it all. And he did, to his credit, but—to my shame—in that moment I did not believe him. I even felt angry, because I could tell that he was already bored with the subject, so bored, in fact, that when I continued to insist on answers he simply ignored me and began to flirt with three tall blonde ladies who had sat down at the table next to us. I got up and went to the toilet.
When I think of the things he did later in life, I despair. Was there any way I could have warmed his heart? Could I have swayed his will, strong as it was, or just planted a tiny seed that might have sprouted and saved some part of his soul? But I did nothing, said nothing, not a word. I did not even try, because I was so afraid for myself. And that, that still pains me, though I’m sure that anything I could have said would have been met with ridicule. One of his tasteless jokes, no doubt. Yes, I can hear him now. A rabbi, a priest, and a horse walk into a bar. Oy vey! How much difference could I have made? We mustn’t forget that it was not just he who played with fire. His entire generation set loose the hounds of hell. Even so, I cannot help but feel guilty, because I was his first teacher. He was a pup when I met him, a child in my care, and the habits we develop in youth are what we follow in old age. So I failed him, failed him miserably in what is most important: I was unable to communicate the sanctity, the holiness of our discipline. I did not teach him what the “pure” in “pure mathematics” really means. It is not what people think. It is not knowledge for its own sake. It is not a search for patterns, nor is it a series of abstract, intellectual games completely unconnected from the real world and its many troubles. It is something quite other. Mathematics is the closest we can come to the mind of Hashem. And so, it should be practiced with reverence, because it has true power, a power that can be easily used for evil, as it is born from a faculty that only we possess, and that the Lord, blessed be He, gave us instead of teeth, claws, or talons, but that is equally dangerous and lethal. Of this, I taught him nothing. Whatever judgment awaits me, I cannot deny that I saw it before anyone else. What he could do. It was so rare and beautiful that to watch him was to weep. Yes, I saw that, but I also saw something else. A sinister, machinelike intelligence that lacked the restraints that bind the rest of us. Why did I remain silent, then? Because he was so superior. To me, to all of us. I felt slightly ashamed in his presence. Belittled and debased. Nothing but a foolish old man with foolish ideas. And now, older still, I recognize that, in spite of his callousness, he was trying to understand this world at its deepest level. He had a burning vision, an inner fire that I never felt, though I have chased after it. Spiritually, he was an ignoramus, yes, but he did have unquestionable faith in logic. Ah, but that type of faith is always dangerous! Especially if it is later betrayed. Nothing should be beyond question. Moses even questioned the Almighty! And while the Lord, blessed be He, does not answer, the questions themselves may save us. Lost faith is worse than no faith at all, because it leaves behind a gaping hole, much like the hollow that the Spirit left when it abandoned this accursed world. But by their very nature, those god-shaped voids demand to be filled with something as precious as that which was lost. The choice of that something—if indeed it is a choice at all—rules the destiny of men.
